What if you were driving a car at 45 miles per hour and you knew that your destination was less than 150 miles away? What inequality could you set up to solve for the number of hours that you have left to travel? After you've solved the inequality, how could you check to make sure that your answer is correct? Once you've completed this Concept, you'll be able to find and verify solutions to inequalities representing scenarios like these.

Guidance

Sometimes Things Are Not Equal

In some cases there are multiple answers to a problem or the situation requires something that is not exactly equal to another value. When a mathematical sentence involves something other than an equal sign, an
inequality
is formed.

Definition:
An
algebraic inequality
is a mathematical sentence connecting an expression to a value, a variable, or another expression with an inequality sign.

Example A

Translate the following into an inequality: Avocados cost $1.59 per pound. How many pounds of avocados can be purchased for less than $7.00?

Solution:
Choose a variable to represent the number of pounds of avocados purchased, say
.

You will be asked to solve this inequality in the exercises

Checking a Solution to an Inequality

Unlike equations, inequalities have more than one solution. However, you can check whether a value, such as
, is
a solution
to an inequality the same way as you would check if it is
the solution
to an equation - by substituting it in and seeing if you get a true algebraic statement.

The following two examples show you how this works.

Example B

Check whether
is a solution set to
.

Solution:

Plug in
, to see if we get a true statement.

Since
gives us a false statement, it is not a solution to the inequality.

Example C

Check whether
is a solution to
.

Solution:

Substitute in
:

For
to be a true statement, we need
or
. Since
, this is a true statement, so
is a solution.

Video Review

Guided Practice

1. Check whether
is a solution to

2. Check whether
is a solution to

Solutions:

1. Substitute in
, to see if it is a solution to

Since 1 is less than 7, we have a true statement, so
is a solution to

2. Check if
is a solution to

Since 7 is not less than 7, this is a false statement. Thus
is not a solution to

Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: Equations and Inequalities
(16:11)

Define
solution
.

What is the difference between an algebraic equation and an algebraic inequality? Give an example of each.

What are the five most common inequality symbols?

In 4 – 7, define the variables and translate the following statements into algebraic equations.

An amount of money is invested at 5% annual interest. The interest earned at the end of the year is greater than or equal to $250.

You buy hamburgers at a fast food restaurant. A hamburger costs $0.49. You have at most $3 to spend. Write an inequality for the number of hamburgers you can buy.

For exercises 8 – 11, check whether the given solution set is the solution set to the corresponding inequality.

In 12-14, find the solution set.

Using the burger and French fries situation from the previous Concept, give three combinations of burgers and fries your family can buy without spending more than $25.00.

Solve the avocado inequality from Example A and check your solution.

On your new job you can be paid in one of two ways. You can either be paid $1000 per month plus 6% commission on total sales or be paid $1200 per month plus 5% commission on sales over $2000. For what amount of sales is the first option better than the second option? Assume there are always sales over $2000.

Mixed Review

Translate into an algebraic equation: 17 less than a number is 65.

Simplify the expression:
.

Rewrite the following without the multiplication sign:
.

The volume of a box without a lid is given by the formula
, where
is a length in inches and
is the volume in cubic inches. What is the volume of the box when
?

Vocabulary

Language:

An algebraic inequality is a mathematical sentence connecting an expression to a value, a variable, or another expression with an inequality sign.

inequality signs

inequality signs

Listed below are the most common inequality signs.
greater thangreater than or equal toless than or equal toless thannot equal to

solution

solution

The value (or multiple values) that make the equation or inequality true.

Image Attributions

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Description

Learn about inequalities and how to check solutions to inequalities.

Learning Objectives

Here you will learn how to read about a real-life situation and write an inequality that represents this situation. You will then solve the inequality and plug the answer back into the inequality to check your work.